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Chapter 2: Problem 7

Name the angle included between sides \(D E\) and DF in \(\triangle D E F\).

Short Answer

Expert verified
The angle included between sides \( DE \) and \( DF \) is \( \angle EDF \).

Step by step solution

01

Identify the Given Elements

In triangle \( \triangle DEF \), we are given two sides, \( DE \) and \( DF \). We need to identify and name the angle that lies between these two sides.
02

Apply the Concept of Included Angle

The included angle is the angle formed between two adjacent sides of a triangle. In \( \triangle DEF \), the sides \( DE \) and \( DF \) meet at point \( D \). Therefore, the included angle is the one that opens at point \( D \).
03

Name the Included Angle

Based on the understanding from geometry, the angle enclosed by sides \( DE \) and \( DF \) is \( \angle EDF \). This is because vertex \( D \) is where they meet, and it is the angle that connects these two sides. Thus, \( \angle EDF \) is the angle we are looking for.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangles
Triangles are fundamental shapes in geometry, consisting of three sides and three angles. They are categorized by their sides and angles, leading to different types such as equilateral, isosceles, and scalene triangles. Understanding triangles involves knowing how their parts relate to each other, and recognizing how the angles contribute to the triangle's properties is key.
A triangle's angles are crucial for determining its overall shape:
  • **Acute Triangle**: All angles are less than 90掳.
  • **Right Triangle**: One angle is exactly 90掳.
  • **Obtuse Triangle**: One angle is greater than 90掳.
When given a triangle, identifying its sides and angles helps in solving geometry problems effectively. Each vertex of a triangle connects two sides, which also means that the way these sides meet will form different angles.
Included Angle
An included angle is formed between two sides of a triangle and is crucial when determining the shape and size of the triangle. It differs from other angles in a triangle because it is specifically the angle where two sides meet directly.
The concept of the included angle is helpful in problems involving:
  • **Determining Triangle Properties**: The measure of the included angle can influence the type of triangle (e.g., acute or obtuse triangle).
  • **Calculating Areas and Other Metrics**: The included angle can be a vital factor when using trigonometric formulas, like the sine rule or the cosine rule in calculations involving sides of triangles.
In the context of \(\triangle DEF\), the included angle at vertex \(D\) is \(\angle EDF\). It connects sides \(DE\) and \(DF\), making it essential for any further geometric calculations involving this triangle.
Adjacent Sides
Adjacent sides in a triangle are two sides that meet at a common point or vertex. Understanding which sides are adjacent helps in identifying angles like the included angle.
In everyday geometry problems, knowing the adjacent sides aids in:
  • **Naming Angles**: Recognizing which sides form an angle helps in correctly identifying and naming the angles in a triangle.
  • **Solving for Unknowns**: Familiarity with adjacent sides is useful when applying rules like the Pythagorean theorem in the case of right triangles or other trigonometric identities.
For \(\triangle DEF\), the sides \(DE\) and \(DF\) are adjacent, as they meet at vertex \(D\). This relationship is a key aspect for understanding the placement and measure of \(\angle EDF\).

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Most popular questions from this chapter

Name the side included between the angles: \(\angle \mathrm{D}\) and \(\angle \mathrm{F}\) in \(\triangle \mathrm{DEF}\).

Name the angle included between sides \(A B\) and \(B C\) in \(\triangle A B C\).

Name the angle included between sides RS and TS in \(\triangle\) RST.

Draw the triangle with the two sides and the included angle and (2) measure the remaining side and angles: \(\mathrm{DE}=2\) inches, \(\mathrm{DF}=1\) inch, \(\angle \mathrm{D}=60^{\circ}\).

Draw the triangle with the two angles and the included side and (2) measure the remaining sides and angle. \(\triangle \mathrm{ABC}\) with \(\angle \mathrm{A}=40^{\circ}, \angle \mathrm{B}=50^{\circ},\) and \(\mathrm{AB}=3\) inches.
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